Easton's theorem for Ramsey and strongly Ramsey cardinals

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V. Gitman and B. Cody, “Easton’s theorem for Ramsey and strongly Ramsey cardinals,” Annals of Pure and Applied Logic, vol. 166, no. 9, pp. 934–952, 2015.

Since the earliest days of set theory, when Cantor put forth the Continuum Hypothesis in 1877, set theorists have been trying to understand the properties of the continuum function dictating the sizes of powersets of cardinals. In 1904, König presented his false proof that the continuum is not an aleph, from which Zermelo derived the primary constraint on the continuum function, the Zermelo-König inequality, that $\alpha<\text{cf}(2^\alpha)$ for any cardinal $\alpha$. In the following years, Jourdain and Housedorff introduced the Generalized Continuum Hypothesis, and in another two decades Gödel showed the consistency of ${\rm GCH}$ by demonstrating that it held in his constructible universe $L$. The full resolution to the question of ${\rm CH}$ in ${\rm ZFC}$ had to wait for Cohen's development of forcing in 1963, which could be used to construct set-theoretic universes with arbitrarily large sizes of the continuum. Gödel's and Cohen's results together finally established the independence of ${\rm CH}$ from ${\rm ZFC}$. A decade later, building on advances in forcing techniques, Easton showed that, assuming ${\rm GCH}$, any class function $F$ on the regular cardinals satisfying $F(\alpha)\leq F(\beta)$ for $\alpha\leq\beta$ and $\alpha<\text{cf}(F(\alpha))$ can be realized as the continuum function in a cofinality preserving forcing extension (missing reference).)

For some simple and other more subtle reasons, the presence of large cardinals in a set-theoretic universe imposes additional constraints on the continuum function, the most obvious of these being that the continuum function must have a closure point at any inaccessible cardinal. Other restrictions arise from large cardinals with strong reflecting properties. For instance, ${\rm GCH}$ cannot fail for the first time at a measurable cardinal, although Levinski showed in [1] that ${\rm GCH}$ can hold for the first time at a measurable cardinal. Supercompact cardinals impose much stronger constraints on the continuum function. If $\kappa$ is supercompact and ${\rm GCH}$ holds below $\kappa$, then it must hold everywhere and, in contrast to Levinski's result, if ${\rm GCH}$ fails for all regular cardinals below $\kappa$, then it must fail for some regular cardinal $\geq\kappa$ [2]. (Interestingly, in the absence of the axiom of choice, the existence of measurable or supercompact cardinals does not impose any of these restrictions on the continuum function [3].) Additionally, certain continuum patterns at a large cardinal can carry increased consistency strength as, for instance, a measurable cardinal $\kappa$ at which ${\rm GCH}$ fails has the consistency strength of a measurable cardinal of Mitchell order $o(\kappa)=\kappa^{++}$ [4]. Some global results are also known concerning sufficient restrictions on the continuum function in universes with large cardinals. Menas showed in [5] that, assuming ${\rm GCH}$, there is a cofinality preserving and supercompact cardinal preserving forcing extension realizing any locally definable (A function $F$ is locally definable if there is a true sentence $\psi$ and a formula $\varphi(x,y)$ such that for all cardinals $\gamma$, if $H_\gamma\models \psi$, then $F$ has a closure point at $\gamma$ and for all $\alpha,\beta<\gamma$, we have $F(\alpha)=\beta\leftrightarrow H_\gamma\models \varphi(\alpha,\beta)$.) function on the regular cardinals obeying the constraints of Easton's theorem, and Friedman and Honzik extended this result to strong cardinals using generalized Sacks forcing [6]. In [7], Cody showed that if ${\rm GCH}$ holds, and if $F$ is any function obeying the constraints of Easton's theorem ($F$ need not be locally definable) such that each Woodin cardinal is closed under $F$, then there is a cofinality preserving forcing extension realizing $F$ to which all Woodin cardinals are preserved.

In this article, we show that, assuming ${\rm GCH}$, if $\kappa$ is a Ramsey or a strongly Ramsey cardinal, then any class function on the regular cardinals having a closure point at $\kappa$ and obeying Easton's constraints is realized as the continuum function in a cofinality preserving forcing extension in which $\kappa$ remains Ramsey or strongly Ramsey respectively. In particular, this extends Levinski's result mentioned earlier to Ramsey and strongly Ramsey cardinals. Strongly Ramsey cardinals, introduced by Gitman in [8], fall in between Ramsey cardinals and measurable cardinals in consistency strength.

Theorem: Assuming ${\rm GCH}$, if $\kappa$ is a Ramsey or a strongly Ramsey cardinal and $F$ is a class function on the regular cardinals having a closure point at $\kappa$ and satisfying $F(\alpha)\leq F(\beta)$ for $\alpha\leq\beta$ and $\alpha<\text{cf}(F(\alpha))$, then there is a cofinality preserving forcing extension in which $\kappa$ remains Ramsey or strongly Ramsey respectively, and $F$ is realized as the continuum function, namely $2^\delta=F(\delta)$ for every regular cardinal $\delta$.

References

  1. J.-P. Levinski, “Filters and large cardinals,” Ann. Pure Appl. Logic, vol. 72, no. 2, pp. 177–212, 1995.
  2. T. Jech, Set theory. Berlin: Springer-Verlag, 2003, p. xiv+769.
  3. A. W. Apter, “On some questions concerning strong compactness,” Arch. Math. Logic, vol. 51, no. 7-8, pp. 819–829, 2012.
  4. M. Gitik, “On measurable cardinals violating the continuum hypothesis,” Ann. Pure Appl. Logic, vol. 63, no. 3, pp. 227–240, 1993.
  5. T. K. Menas, “Consistency results concerning supercompactness,” Trans. Amer. Math. Soc., vol. 223, pp. 61–91, 1976.
  6. S.-D. Friedman and R. Honzik, “Easton’s theorem and large cardinals,” Ann. Pure Appl. Logic, vol. 154, no. 3, pp. 191–208, 2008.
  7. B. Cody, “Some results on large cardinals and the continuum function,” PhD thesis, The Graduate Center of the City University of New York, 2012.
  8. V. Gitman, “Ramsey-like cardinals,” The Journal of Symbolic Logic, vol. 76, no. 2, pp. 519–540, 2011.